The set of rational numbers, Q, is the set of numbers
formed from the ratio of two
integers, where the numerator can be any integer and the denominator can be any
nonzero
integer. Formally, this definition can be given in set-builder notation as
follows.
Any rational number has an infinite number of
representations, as the fol lowing theorem
illustrates.
Theorem
Let
be any rational number and n any nonzero integer. Then
 |
To de termine the equality of two rational numbers, we can
use the following definition.
Definition
Let
and
be any rational numbers. Then
if and only if ad = bc . |
We can re present any rational number on a number line .
Addition
Let
and
be any rational numbers. Then
 |
It is convention to express the sum , difference , product ,
or quotient of two rational
numbers in its simplest form .
To ensure the denominator is positive , the following
theorem may be applied.
Theorem
Let
be any rational number. Then
 |
Properties of Rational Number Addition
Let
be any rational numbers.
• Closure:
is a rational number.
• Commutative:
• Associative:
• Identity:
, where 0 can be represented as 0/m , for m ≠ 0
• Additive Inverse :
Theorem: Additive Cancellation
Let
and
be any rational numbers. If
 |
Theorem: Opposite of the Opposite
Let
be any rational number. Then
 |
Subtraction
Let
and
be any rational numbers. Then
 |
We can view rational number subtraction as adding the
opposite, as the following
definition illustrates.
Definition
Let
and
be any rational numbers. Then
 |
and
be any rational numbers. Then
